Standard quantum teleportation

Today, quantum teleportation [1] is recognised as one of the pillars of quantum information theory. In addition to its foundational value, quantum teleportation is used as a building block for several core ideas in quantum computation [2,3,4], quantum communication [5,6,7], and quantum information [8,9,10]. In a nutshell, quantum teleportation is a protocol that allows distant parties to transfer quantum information over a classical channel at the cost of consuming entanglement.

Let us say that Alice and Bob share a maximally entangled qubit pair $\vert\phi^+\rangle:=\frac{\vert{00}\rangle+\vert{11}\rangle}{\sqrt{2}}$ and Alice wants to teleport an arbitrary qubit $\vert\psi\rangle$. The standard quantum teleportation protocol can be described in three steps:

1. Joint measurement: Alice performs a Bell measurement on the target state $\vert\psi\rangle$ she wants to teleport and her part of the maximally entangled state. Depending on the result of Alice’s measurement, the state held by Bob is transformed into one of the four states $\Big\{\vert\psi\rangle, \; \sigma_X\vert\psi\rangle, \;\sigma_Y\vert\psi\rangle, \;\sigma_Z\vert\psi\rangle\Big\}$, where $\sigma_X, \sigma_Y, \sigma_Z$ are the Pauli matrices.
2. Classical communication: Alice sends Bob the classical outcome obtained in the joint measurement.
3. Correction: Bob performs a Pauli operation corresponding to Alice’s measurement outcome on his system to ensure he obtains exactly $\vert\psi\rangle$.

In one out of four cases, the state held by Bob is exactly the target state $\vert\psi\rangle$, but due to the intrinsic probabilistic nature of quantum measurements, with probability $3/4$, a non-trivial correction is required for Bob to recover $\vert\psi\rangle$. In addition to being cumbersome and adding complexity to the protocol, the correction step is a strong limiting factor for some applications of quantum teleportation methods in other protocols. In particular, the correction step prevents the possibility of deterministic exact universal programming and gate teleportation [2,3], protocols which are highly desired for quantum computation purposes. Could we avoid or simplify this correction step?

PBT: A teleportation protocol which does not require the corrections

In the seminal papers [11,12], Ishizaka and Hiroshima presented a way to greatly simplify the correction step of the standard quantum teleportation in a scheme now known as $\textit{port-based teleportation}$ (PBT). One key idea of PBT is to allow more copies of the resource state. Instead of sharing a single copy of the maximally entangled state, in PBT Alice and Bob share $N$ copies of a maximally entangled pair. This way, Bob has access to $N$ different subsystems and each of these subsystems is identified as a port to which the target state may be teleported.

Port-based teleportation comes in two variants, deterministic or probabilistic. The deterministic PBT can be described in three steps. First, Alice performs a joint measurement on the state $\vert\psi\rangle$ she wants to teleport and her part of the shared resource state $\vert\phi^+\rangle^{\otimes N}$. This joint measurement has $N$ outcomes, each one corresponding to a different port. Second, Alice sends the outcome of her measurement to Bob. Third, Bob finds the (mixed) state $\rho$ in the port corresponding to Alice’s outcome and discards all other subsystems. In other words, the correction required by Bob is simply to select a particular subsystem.
In deterministic PBT, the final state $\rho$ held by Bob is not identical to the target state $\vert\psi\rangle$ and depends on the measurement performed by Alice, which may be optimised in order to maximise its performance. The performance is evaluated by the fidelity between Bob’s output state $\rho$ uniformly averaged over all pure target states $\vert\psi\rangle$, a quantity mathematically described by $f=\int \langle\psi \vert \rho\vert\psi\rangle\,\text{d}{\vert\psi\rangle}$. For the qubit case, Ref. [11] presents a closed expression for the maximal average fidelity which approaches one and behaves as $f\approx 1-\frac{1}{N}$ for large $N$.

In the probabilistic version, Alice performs a joint measurement with $N+1$ outcomes where $N$ of these outcomes correspond to the $N$ ports and the extra outcome corresponds to failure. When the failure outcome is obtained, the target state $\vert\psi\rangle$ is lost and the protocol is aborted. However, if a non-failure outcome is obtained, the target state $\vert\psi\rangle$ is perfectly transferred to its corresponding port. For the qubit case, Ref. [12] presents a closed expression for the maximal probability of success which approaches one and behaves as $p_s\approx 1-\frac{1}{\sqrt{N}}$ for large $N$.

Interestingly, apart from the $N=1$ case, the maximally entangled state is not the optimal resource state for PBT. That is, one can improve the performance of PBT by optimising over the resource state $\vert\phi_\text{PBT}\rangle \in \mathbb{C}_d^{\otimes 2N}$ shared by Alice and Bob. For qubits, optimising over the resource state offers a quadratic improvement over strategies employing the maximally entangled state [12]. That is, for the optimal state, the average fidelity behaves as $f\approx 1-\frac{1}{N^2}$ for the deterministic case, and the optimal success probability behaves as $p\approx 1-\frac{1}{N}$ for the probabilistic case.

The mathematical methods behind PBT involve the representation theory of the special unitary group $\text{SU}(d)$ and the analysis for the qubit case is greatly simplified by the existence of closed expressions for the irreducible representations of $\text{SU}(2)$. However, the performance of PBT for teleporting qudits remained an open question for several years, until Refs. [13,14,15,16] developed mathematical tools for tackling PBT in its full complexity. In particular, Refs. [13,14] present the best performance for deterministic and probabilistic PBT in any dimension $d$ for the case of the optimal resource state and for $N$ maximally entangled states.

Thanks to its rich mathematical structure, studies in PBT led to results beyond its initial motivations and found applications such as the identification of causal-effect relations [17], transposing and inverting unitary operations [18,19], storage and retrieval of unitary channels [20], communication complexity and Bell nonlocality [21], quantum channel discrimination [22], $\textit{etc}$.

However, all teleportation protocols described above consider the teleportation of a single system. How should one proceed to teleport a quantum state which has a $k$-subsystem structure?

Multi-port-based teleportation

A straightforward approach to teleport a quantum state composed of $k$ subsystems is to consider them as a single system and perform standard PBT in a larger space. Another method would be to perform $k$ independent instances of the standard PBT, a technique which may even be improved by an adaptive scheme that “recycles” the resource state after each round [23]. But none of these approaches is optimal. The best strategy consists of exploiting the subsystem structure of the target state to implement a multi-PBT (MPBT) protocol [24,25].

The core difference between PBT and MPBT is that instead of teleporting the target state into a single port, each subsystem of the target state is teleported into a different port. While Bob discards $N-1$ subsystems after receiving Alice’s outcome in standard PBT, the correction made by Bob in MPBT consists of discarding $N-k$ subsystems, and, after this discarding process, Bob must identify which of the remaining ports corresponds to each subsystem of the target state.
Hence, in MPBT, after performing her joint measurement Alice should inform Bob not only what to discard, but also the order in which the remaining ports should be arranged.

References [24,25] identified all the symmetries involved in MPBT and developed the mathematical tools to tackle MPBT. In particular, they provided a good characterisation for the set of linear operators commuting with $U^{\otimes N}\otimes \overline{U}^{\otimes k}$ where $U$ is any $d$-dimensional unitary operator and $\overline{U}$ its complex conjugate. This mathematical formulation allowed the authors to solve the scenario where Alice and Bob share $N$ copies of a maximally entangled state. In particular, Ref. [24] found optimal joint measurements for deterministic and probabilistic PBT in any dimension $d$, for any number of subsystems $k$, and any number $N$ of copies of maximally entangled qudit pairs. Nevertheless, results for the scenario with optimal resource states were still lacking.

In a paper recently published in $\textit{Quantum}$ [26], Mozrzymas, Studzi\’n{}ski, and Kopszak conclude a sequence of results on MPBT by providing a complete solution for the scenario where the resource state is optimised. By sharpening the group representation theory methods used for MPBT, Ref. [26] was able to obtain the optimal resource state and optimal measurements for probabilistic and deterministic MPBT, completing a chapter of our knowledge about teleportation protocols. Remarkably, the optimal performance for optimal probabilistic MPBT admits a surprisingly simple formula. The averaged probability of success for teleporting $k$ subsystems of dimension $d$ with the optimal probabilistic MPBT with $N$ ports is $p_s=\binom{N}{k}\binom{N+d^2-1}{k}^{-1}$.

Over the years, we have been investigating quantum teleportation protocols and adapting to particular requirements and situations. Taming MPBT represents substantial progress in this investigation. Moreover, the mathematical methods employed for solving MPBT have a strong potential to find applications in fields which go beyond teleportation and its immediate consequences.

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